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    "using ApproxFun, Plots, ComplexPhasePortrait, ApproxFun,  SingularIntegralEquations,\n",
    "        SpecialFunctions, DualNumbers, SO\n",
    "\n",
    "using SingularIntegralEquations.HypergeometricFunctions\n",
    "gr();"
   ]
  },
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   "cell_type": "markdown",
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   "source": [
    "# M3M6: Methods of Mathematical Physics\n",
    "\n",
    "$$\n",
    "\\def\\dashint{{\\int\\!\\!\\!\\!\\!\\!-\\,}}\n",
    "\\def\\infdashint{\\dashint_{\\!\\!\\!-\\infty}^{\\,\\infty}}\n",
    "\\def\\D{\\,{\\rm d}}\n",
    "\\def\\E{{\\rm e}}\n",
    "\\def\\dx{\\D x}\n",
    "\\def\\dt{\\D t}\n",
    "\\def\\dz{\\D z}\n",
    "\\def\\ds{\\D s}\n",
    "\\def\\C{{\\mathbb C}}\n",
    "\\def\\R{{\\mathbb R}}\n",
    "\\def\\H{{\\mathbb H}}\n",
    "\\def\\CC{{\\cal C}}\n",
    "\\def\\HH{{\\cal H}}\n",
    "\\def\\FF{{\\cal F}}\n",
    "\\def\\I{{\\rm i}}\n",
    "\\def\\Ei{{\\rm Ei}\\,}\n",
    "\\def\\qqqquad{\\qquad\\qquad}\n",
    "\\def\\qqand{\\qquad\\hbox{and}\\qquad}\n",
    "\\def\\qqfor{\\qquad\\hbox{for}\\qquad}\n",
    "\\def\\qqwhere{\\qquad\\hbox{where}\\qquad}\n",
    "\\def\\Res_#1{\\underset{#1}{\\rm Res}}\\,\n",
    "\\def\\sech{{\\rm sech}\\,}\n",
    "\\def\\acos{\\,{\\rm acos}\\,}\n",
    "\\def\\erfc{\\,{\\rm erfc}\\,}\n",
    "\\def\\vc#1{{\\mathbf #1}}\n",
    "\\def\\ip<#1,#2>{\\left\\langle#1,#2\\right\\rangle}\n",
    "\\def\\br[#1]{\\left[#1\\right]}\n",
    "\\def\\norm#1{\\left\\|#1\\right\\|}\n",
    "\\def\\half{{1 \\over 2}}\n",
    "\\def\\fL{f_{\\rm L}}\n",
    "\\def\\fR{f_{\\rm R}}\n",
    "\\def\\HF(#1,#2,#3,#4){{}_2F_1\\!\\left({#1,#2 \\atop #3}; #4 \\right)}\n",
    "\\def\\questionequals{= \\!\\!\\!\\!\\!\\!{\\scriptstyle ? \\atop }\\,\\,\\,}\n",
    "$$\n",
    "\n",
    "Dr Sheehan Olver\n",
    "<br>\n",
    "s.olver@imperial.ac.uk\n",
    "\n",
    "Office Hours: 3-4pm Mondays, 11-12am Thursdays, Huxley 6M40\n",
    "<br>\n",
    "Website: https://github.com/dlfivefifty/M3M6LectureNotes\n"
   ]
  },
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    "\n",
    "# Lecture 27:  Singular points of ordinary differential equations\n",
    "\n",
    "\n",
    "2. Singular points of ODEs\n",
    "   - regular singular point\n",
    "   - irregular singular point\n",
    "   - singular points at ∞"
   ]
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   "cell_type": "markdown",
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   "source": [
    "## Singular points\n",
    "\n",
    "We will consider the first order ODE\n",
    "$$\n",
    "u' + a(z) u = 0\n",
    "$$\n",
    "and the second order ODE\n",
    "$$\n",
    "u'' + a(z) u' + b(z) u = 0\n",
    "$$\n",
    "where $a$ and $b$ are rational.  \n",
    "\n",
    "The simplest point is an ordinary point, where we know that $u$ is analytic:\n",
    "\n",
    "**Definition (ordinary point)** $z_0$ is a ordinary point of the ODE if $a(z)$ and $b(z)$ are analytic at $z_0$.\n",
    "\n",
    "\n",
    "### Regular singular point\n",
    "\n",
    "Suppose $a$ has a simple pole, and w.l.o.g. take $z_0=0$, that is: \n",
    "$$\n",
    "a(z) = {a_{-1} \\over z} + a_0 + a_1 z + a_2 z^2 + \\cdots \n",
    "$$\n",
    "where $a_0 + a_1 z + a_2 z^2 + \\cdots $ is analytic.\n",
    "Take as an ansatz to the first order ODE\n",
    "$$u(z) = z^\\alpha  ( 1 + z v(z)) = z^\\alpha + z^{\\alpha+1} v$$\n",
    "where $\\alpha = a_{-1}$. Note that $c u(z)$ is a solution of $u'(z) = a(z) u(z)$: the choice of $1$ above makes the solution unique. \n",
    "\n",
    "\n",
    "Note that\n",
    "$$\n",
    "u'(z) = \\alpha z^{\\alpha-1} + z^\\alpha ((\\alpha+1) v + z v')\n",
    "$$\n",
    "The ODE $u'(z) = a(z) u(z)$ becomes an inhomogenous equation:\n",
    "\\begin{align*}\n",
    "z^\\alpha ((\\alpha+1) v + z v' - z a(z) v) &= z^{\\alpha} (a(z) - {\\alpha \\over z}  ) & \\Leftrightarrow\\\\\n",
    "z v'(z)  + (\\alpha+1 - z a(z)) v(z) &= a_0 + a_1 z + a_2 z^2 + \\cdots\n",
    "\\end{align*}\n",
    "In operator form, expressing \n",
    "$$\n",
    "v(z) = v_0 + v_1z + v_2 z^2 = (1,z,z^2,\\ldots)\\begin{pmatrix}v_0\\\\v_1\\\\\\vdots \\end{pmatrix}\n",
    "$$\n",
    "we have\n",
    "\\begin{align*}\n",
    "z v' &=(1,z,z^2,\\ldots)  \\begin{pmatrix} 0 \\cr & 1 \\cr && 2 \\cr &&& 3 \\cr &&&&\\ddots \\end{pmatrix} \\begin{pmatrix}v_0\\\\v_1\\\\\\vdots \\end{pmatrix} \\\\\n",
    "(\\alpha+1 - z a(z)) v(z) &= (1,z,z^2,\\ldots)  \\begin{pmatrix} 1 \\cr -a_0 & 1 \\cr -a_1 & -a_0& 1 \\cr -a_2 & -a_1&-a_0& 1 \\cr \\vdots&\\ddots&\\ddots&\\ddots&\\ddots \\end{pmatrix} \\begin{pmatrix}v_0\\\\v_1\\\\\\vdots \\end{pmatrix}\n",
    "\\end{align*}\n",
    "Thus the equation for our unknown coefficients becomes:\n",
    "$$\n",
    "\\begin{pmatrix} 1 \\cr -a_0 & 2 \\cr -a_1 & -a_0& 3 \\cr -a_2 & -a_1&-a_0& 4 \\cr \\vdots&\\ddots&\\ddots&\\ddots&\\ddots \\end{pmatrix} \\begin{pmatrix}v_0\\\\v_1\\\\\\vdots \\end{pmatrix} = \\begin{pmatrix}a_0\\\\a_1\\\\\\vdots \\end{pmatrix}\n",
    "$$\n",
    "This equation induces decay in $|v_k|$ just like last lecture, hence we know its analytic with the same radius of convergence as $a$.\n",
    "\n",
    "Here is an example: as we can see, `v(z)` is analytic in a disk of radius `r` where `r` is dictated by the function `a`. "
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       "</g>\n",
       "</svg>\n"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "r = 0.9  # radius we are solving in, can be up to but not including 1\n",
    "S = Taylor(Circle(r))\n",
    "α = 0.3\n",
    "a = z -> α/z + 1/(1-z)\n",
    "ã = Fun( z -> a(z) - α/z, S)\n",
    "za = Fun( z -> z*a(z), S)\n",
    "z = Fun(S)\n",
    "D = Derivative() : S → S\n",
    "L = z*D + (α+1-za)\n",
    "v = L \\ ã\n",
    "\n",
    "# v is analytic in disk of radius r\n",
    "phaseplot(v, (-1,1), (-1,1))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here is a plot of `u`, which shows that it has a branch cut:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
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       "\" transform=\"translate(39, 12)\"/>\n",
       "</g>\n",
       "</svg>\n"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# u has a branch \n",
    "u = z -> z^α + z^(α+1)*v(z)\n",
    "phaseplot(u, (-1,1), (-1,1))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can confirm it solves the ODE:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4.884981308350689e-15 - 7.521914234669674e-17im"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "up = dualpart(u(dual(0.1,1.0))) # uses DualNumbers.jl to calculate derivative\n",
    "\n",
    "up - a(0.1)u(0.1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Definition (first order regular singular point)** $z_0$ is a regular singular point of the first order ODE if $(z-z_0)a(z)$ is analytic at $z_0$.\n",
    "\n",
    "For second order equations, we will arrive at the following analogue:\n",
    "\n",
    "**Definition (second order regular singular point)** $z_0$ is a regular singular point of the second order ODE if $(z-z_0) a(z)$ and $(z-z_0)^2b(z)$ are analytic at $z_0$.\n",
    "\n",
    "Assume \n",
    "$$\n",
    "a(z) = {a_{-1} \\over z} + a_0 + a_1 z + \\cdots\n",
    "$$\n",
    "and\n",
    "$$\n",
    "b(z) = {b_{-2} \\over z^2} + {b_{-1} \\over z} + b_0 + b_1 z + \\cdots\n",
    "$$\n",
    "To determine the right ansatz, consider\n",
    "$$\n",
    "u'' + {a_{-1} \\over z} u' + {b_{-2} \\over z^2} u'' = 0\n",
    "$$\n",
    "which has solutions $z^\\alpha$ solving the indicial equation:\n",
    "\n",
    "**Definition (indicial equation)**\n",
    "$$\n",
    "\\alpha(\\alpha-1) + a_{-1} \\alpha + b_{-2} = 0\n",
    "$$\n",
    "\n",
    "Now take as an ansatz:\n",
    "$$\n",
    " u(z) = z^\\alpha (1 + z v(z)) = z^\\alpha + z^{\\alpha+1} v(z)\n",
    "$$\n",
    "Differentiating we have\n",
    "\\begin{align*}\n",
    "u'(z) &= \\alpha z^{\\alpha-1} + z^\\alpha((\\alpha+1) v + z v') \\\\\n",
    "u''(z) &= \\alpha(\\alpha-1) z^{\\alpha-2} + z^{\\alpha-1}( \\alpha(\\alpha+1) v + \\alpha z v' + (\\alpha+1) z v' + z v' + z^2 v''  ) \\\\\n",
    " &= \\alpha(\\alpha-1) z^{\\alpha-2} + z^{\\alpha-1}( \\alpha(\\alpha+1) v  + 2(\\alpha+1) z v'  + z^2 v''  ) \n",
    "\\end{align*}\n",
    "Thus $v$ solves \n",
    "$$\n",
    "z^2 v''+ z (2(\\alpha+1)  + z a(z)) v' + ( \\alpha(\\alpha+1) +  z a(z) (\\alpha+1) + z^2 b(z)) v    = - {\\alpha(\\alpha-1) \\over z} - a(z) \\alpha -z b(z) = \\underbrace{- \\alpha a_0 - b_{-1} - (\\alpha a_1 + b_0) z - (\\alpha a_2 + b_1) z^2 + \\cdots}_{f(z) = f_0 + f_1 z + \\cdots}\n",
    "$$\n",
    "\n",
    "The question is when can forward substitution proceed.  Let's write out the operators, first we have:\n",
    "$$\n",
    "z^2 v'' \\equiv \\begin{pmatrix} 0 \\cr & 0 \\cr &&2 \\cr &&& 6 \\cr &&&&12 \\cr &&&&&\\ddots \\end{pmatrix}\n",
    "$$\n",
    "Then we have\n",
    "$$\n",
    "2 (\\alpha+1) z  v' \\equiv \\begin{pmatrix} 0\\cr & 2 (\\alpha+1)  \\cr &&&4 (\\alpha+1) \\cr &&&& 6(\\alpha+1) \\cr &&&&&8 (\\alpha+1) \\cr &&&&&&\\ddots \\end{pmatrix}\n",
    "$$\n",
    "and\n",
    "$$\n",
    "z^2 a(z) v' = (z a(z))(z v')  \\equiv  \\begin{pmatrix} a_{-1} \\\\ a_0 & a_{-1} \\\\ a_1 & a_0 & a_{-1} \\\\  a_2 & a_1 & a_0 & a_{-1} \\\\ \\vdots &\\ddots&\\ddots&\\ddots&\\ddots \\end{pmatrix} \\begin{pmatrix} 0 \\cr & 1   \\cr &&2  \\cr &&& 3 \\cr &&&&4  \\cr &&&&&\\ddots \\end{pmatrix}\n",
    "$$\n",
    "Finally since\n",
    "\\begin{align*}\n",
    "\\alpha(\\alpha+1) + z (\\alpha+1) a(z) + z^2 b(z) &= \\alpha(\\alpha-1) +  \\alpha a_{-1} +  b_{-2} + 2 \\alpha + a_{-1} + ((\\alpha+1) a_0 + b_{-1}) z + ((\\alpha+2) a_1 + b_0) z^2 + \\cdots \\\\\n",
    "& = 2 \\alpha + a_{-1} + ((\\alpha+1) a_0 + b_{-1}) z + ((\\alpha+2) a_1 + b_0) z^2 + \\cdots\n",
    "\\end{align*}\n",
    "we have\n",
    "$$\n",
    "(\\alpha(\\alpha+1) + z (\\alpha+1) a(z) + z^2 b(z) ) v \\equiv \n",
    "\\alpha \\begin{pmatrix} \n",
    "2 \\alpha + a_{-1} \\\\ \n",
    "(\\alpha+1) a_0 + b_{-1} & 2 \\alpha + a_{-1} \\\\ \n",
    "(\\alpha+1) a_1 + b_0 & (\\alpha+1) a_0 + b_{-1} & 2 \\alpha + a_{-1} \\\\ \n",
    "(\\alpha+1) a_2 + b_1 & (\\alpha+1) a_1 + b_0 & (\\alpha+1) a_0 + b_{-1} & 2 \\alpha + a_{-1} \\\\ \n",
    "\\vdots &\\ddots&\\ddots&\\ddots&\\ddots \\end{pmatrix}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We need to only worry about the diagonal to determine if forward substitution proceeds, in this case we have\n",
    "$$\n",
    "\\begin{pmatrix}\n",
    "2 \\alpha + a_{-1} \\\\\n",
    "\\times & 4 \\alpha + 2  + 2a_{-1} \\\\\n",
    "\\times & \\times &6 \\alpha   + 3a_{-1} + 6 \\\\\n",
    "\\times & \\times & \\times & 8 \\alpha  + 4a_{-1} + 12 \\\\\n",
    "\\times & \\times & \\times & \\times & 10 \\alpha  + 5a_{-1} + 20\\\\\n",
    "\\vdots & \\ddots & \\ddots & \\ddots & \\ddots & \\ddots\n",
    "\\end{pmatrix}\n",
    "$$\n",
    "Thus the first condition for this to give us a solution is that $2 (k+1)\\alpha + (k+1)a_{-1} + k(k+1)  \\neq 0$ for all $k=1,2,\\ldots$.\n",
    "This can be thought of another way: note that plugging $\\alpha+k+1$ in to the indicial equation gives us:\n",
    "$$\n",
    "(\\alpha+k+1) (\\alpha+k) + a_{-1} (\\alpha+k+1) + b_{-2} = (\\alpha) (\\alpha-1) + a_{-1} \\alpha + b_{-2}+ 2 (k+1)\\alpha + (k+1)a_{-1} + k(k+1)  = 2 (k+1)\\alpha + (k+1)a_{-1} + k(k+1) \n",
    "$$\n",
    "Thus forward substitution fails if and only $\\alpha+k+1$ is the second root to the indicial equation for some $k=0,1,2,\\ldots$.\n",
    "\n",
    "**Remark**  If this condition is satisfied, then $v$ is analytic near the singular point, which follows by bounding the forward elimination as before, but we'll skip this for now.\n",
    "\n",
    "**Remark** If $\\alpha+k+1$ is an integer, then using the ansatz $z^{\\alpha+k+1}(1 + z v(z))$ proceeds without problem. To get the second solution, we need to introduce logarithms, but we'll skip this for now."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We now check the results of solving\n",
    "$$\n",
    "z^2 v''+ z (2(\\alpha+1)  + z a(z)) v' + ( \\alpha(\\alpha+1) +  z a(z) (\\alpha+1) + z^2 b(z)) v    = - {\\alpha(\\alpha-1) \\over z} - a(z) \\alpha -z b(z)\n",
    "$$\n",
    "numerically.\n",
    "\n",
    "We first choose a solutino to the an indicial equation:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "α = 1.3830951894845303\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "2.7755575615628914e-16"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "S = Taylor()\n",
    "z = Fun(S)\n",
    "D = Derivative() : S\n",
    "\n",
    "a₋₁ = -0.6\n",
    "b₋₂ = 0.3\n",
    "\n",
    "a = z -> a₋₁/z + 1/(2-z)\n",
    "ã = Fun( z -> a(z) -  a₋₁/z, S)\n",
    "za = Fun( z -> z*a(z), S)\n",
    "\n",
    "b = z -> b₋₂/z^2 + 0.4/z + 1/(3-z)\n",
    "zb̃ = Fun(z -> z*b(z) - b₋₂/z, S)\n",
    "z²b = Fun( z -> z^2*b(z), S)\n",
    "\n",
    "α = ((1-a₋₁) +  sqrt((a₋₁-1)^2-4b₋₂))/2\n",
    "@show α\n",
    "α*(α-1) + α*a₋₁ + b₋₂"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can solve for `v`, which is analytic in a neighbourhood of 0:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
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      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "L = z^2*D^2 + z*(2(α+1) + za)*D + (α*(α+1) + za*(α+1) + z²b)\n",
    "v = L \\ (-α*ã - zb̃)\n",
    "\n",
    "phaseplot(v, (-3,3), (-3,3))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Thus one solution is:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
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       "RK5CYII=\n",
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       "</g>\n",
       "</svg>\n"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "u = z -> z^α + z^(α+1)*v(z)\n",
    "\n",
    "phaseplot(u, (-3,3), (-3,3))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's double check it satisfies the original ODE:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3.3654295491028256e-12"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# dual numbers don't work to second degree so let's check numerically: \n",
    "uf = Fun(u, 0.1 .. 0.2)\n",
    "norm(uf'' + Fun(a, space(uf))*uf' + Fun(b, space(uf))*uf)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The second, linearly independent solution can be found taking the other $\\alpha$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "α̃ = 0.21690481051546984\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "1.6653345369377348e-16"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "α̃ = ((1-a₋₁) -  sqrt((a₋₁-1)^2-4b₋₂))/2\n",
    "@show α̃\n",
    "α̃*(α̃-1) + α̃*a₋₁ + b₋₂"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
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      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "L = z^2*D^2 + z*(2(α̃+1) + za)*D + (α̃*(α̃+1) + za*(α̃+1) + z²b)\n",
    "ṽ = L \\ (-α̃*ã - zb̃)\n",
    "\n",
    "phaseplot(ṽ, (-3,3), (-3,3))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
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      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ũ = z -> z^α + z^(α+1)*ṽ(z)\n",
    "\n",
    "phaseplot(ũ, (-3,3), (-3,3))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Any general solution to the ODE is a linear combination of these two."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "### Irregular singular points\n",
    "\n",
    "Every isolated singularity that is not regular is an _irregular singular point_: even something as simple as ${\\D u / \\dz} = {u \\over z^2}$ has an irregular singular point.  Irregular singular points correspond to essential singularities: in this case the solution is\n",
    "$$\n",
    "u(z) =  \\E^{-1/z}\n",
    "$$\n",
    "\n",
    "### Singular points at infinity\n",
    "\n",
    "To determine the type of singularity at $\\infty$ we do our usual change of variables $u(z) = v(1/z)$. For example, if\n",
    "$$\n",
    "{\\D u \\over \\dz} = a(z) u\n",
    "$$\n",
    "Then \n",
    "$$\n",
    "{\\D v \\over \\dz} = -z^{-2}a(z^{-1}) v\n",
    "$$\n",
    "**Definition (first order ordinary point at ∞)** $\\infty$ is an ordinary point of a first order ODE if $z^2a(z)$ is analytic at $\\infty$. \n",
    "\n",
    "**Definition (first order regular singular point at ∞)** $\\infty$ is a regular singular point of a first order ODE if $za(z) $ is analytic at $\\infty$. \n",
    "\n",
    "For second order, we see that $v$ satisfies\n",
    "\\begin{align*}\n",
    "v'(z) &= -z^{-2} u'(z^{-1}) \\\\\n",
    "v''(z) &=2 z^{-3} u'(z^{-1}) + z^{-4} u''(z^{-1}) =-2 z^{-1} v'(z)+ z^{-4} u''(z^{-1})\n",
    "\\end{align*}\n",
    "which gives us the ODE\n",
    "$$\n",
    "u'' + a(z) u' + b(z) u \\Rightarrow\n",
    "z^4v'' + (2 z^3 - a z^2) v'  + b v\n",
    "$$\n",
    "\n",
    "\n",
    "**Definition (second order regular singular point at ∞)** $\\infty$ is a regular singular point of a first order ODE if $za(z)$ and $z^2 b(z)$ are analytic at $\\infty$. \n",
    "\n",
    "\n"
   ]
  }
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